Continuous and coherent texture mapping on deforming objects
If you want the texture to deform with the surface, you will have to use UV mapping. Every polygon corner gets assigned a location in UV (2D) space and textures use this information to map to surface. This mapping is not hard to create (see UV textures/ing tutorials), but it can be unique for every mesh, so the key is ensuring this UV mapping is the same in all your meshes.

There is a way to have only 1 mesh and to use all the others for creating mesh shape keys. You need Absolute shape keys that animate in a sequence. You do this by joining 2 meshes with Join as Shapes option. This way you can convert multiple meshes with the same vertex order into a single mesh with deformation animation.

In python with both selected and the target for animation active:

bpy.ops.object.join_shapes()


You might need to override the context.